How to sync an app progress across devices?

My wife keeps insisting that she has to turn off all Apple devices (iphone, ipad, ipad mini) in order to sync an app progress. I don't think she needs to, but she doesn't believe me. For example, an iPhone shows the game level to be 185 (Frozen Free Fall), while she's playing on an ipad with level 187. She wants the non-playing devices to sync as well.

All the devices are using the same iTunes AppleID, while the Game Center nickname / profile is the same for iPhone and iPad but different for iPad mini. We wanted to keep scores, achievements, and progress separate according to the Game Center ID.

So is she correct for turning off and on the devices? How do you force to sync manually so the update progress is shown on the non-playing device?

2. Can one AppleID have more than one Game Center ID?

3. Some apps give you the option to log in to Facebook. I notice that it tracks and updates the progress when switching to another device if you log in to Facebook on that device. Then is Facebook given priority over the Game Center ID regarding game progress? This is what is confusing to me.

4. Not all games appear in Game Center, e.g. Frozen Free Fall. Why is that?

5. In Game Center section of Settings app, does the AppleID name have to be the same as Game Center nickname? This one is different than the AppleID used in iTunes and iCloud, right? The term "AppleID" is confusing, given that it is used for different purposes.

6. How do I stop populating ALL the contacts from the iPhone to the ipad and ipad mini? We don't make calls on iPad and iPad mini. If I delete the contact on the iPad and iPad mini, it will delete the same contact on the iPhone as well, and we don't want that.

Thanks for responding.